TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Spherical mean-value property for harmonic functions

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

If $u \in C^{2} (Ω)$ and $Δ u = 0$ , then $u (x) = M_{u} (x, r)$ whenever $B_{r} (x) ⋐ Ω$ .

Proof

Given: Countable Choice, $u$ is classically harmonic and $B_{r} (x) ⋐ Ω$ .

1.1

Radial derivative of a spherical average gives $\frac{d}{d t} M_{u} (x, t) = 0$ for $0 < t \leq r$ [given].

1.2

Hence $M_{u} (x, t)$ is constant, and continuity gives $\lim_{t ↓ 0} M_{u} (x, t) = u (x)$ [given].

2.1

Its value at $r$ is therefore $u (x)$ [step 1.2]. ∎

Depends on

Used by

Dependency tree · two levels

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Sources

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